Since phenol and water are immiscible, two phases form: a water (a.k.a. aqueous) phase and a phenol phase. Phenol is the more dense of the two liquids so it sits on the bottom.
The phases are then mixed thoroughly. This forces the phenol into the water layer where it forms an emulsion of droplets throughout. The proteins in the water phase are denatured and partition into the phenol, while the DNA stays in the water.
Then, the mixture is centrifuged and the phases separate. The DNA containing water phase can now be pipetted off and the phenol/protein solution is discarded. Commonly, the DNA is then de-salted and concentrated using ethanol precipitation.
Water is a very polar solvent because the oxygen atom is very electronegative so it “sucks” the electrons towards it and away from the hydrogens, creating a slight negative charge on the oxygen and a slight positive on the hydrogens. i.e. the charge is “polarised” within the molecule.
Phenol is a less polar molecule than water. Although it has a highly electronegative oxygen, this is counteracted by the phenyl ring, which is also very electronegative so there is no concentration of electron density around the oxygen. i.e. the charge is not so polarised in a phenol molecule.
The solubility of the proteins is flipped by phenol
But, proteins are a different story entirely.
As you know proteins are made up of long chains of amino acids. Each amino acid has it’s own characteristics, due to the nature of their side chains. Some, (e.g. phenylalanine, leucine, and tryptophan) are non-polar, because their side chains contain no charged entities. Conversely, amino acids with side chains containing charged entities (e.g. glutamate, lysine and histidine) are polar.